For a permutation $z$ in the symmetric group $\mathrm{S}_{n}$, denote by $L_{z}$ the corresponding simple highest weight module in the principal block of the BGG category $\mathcal{O}$ for the Lie algebra $\mathfrak{sl}_{n}(\mathbb{C})$. In this paper, we provide a combinatorial answer to Kostant's problem for the modules $L_{z}$ when $z$ has shape (associated Young diagram/integer partition via Robinson-Schensted correspondence) equal to $(n-2,1,1)$ or $(n-3,2,1)$. Moreover, we verify that certain closely related conjectures hold for such permutations, including the Indecomposability Conjecture, which states that applying any indecomposable projective functor to the corresponding simple highest weight module outputs either an indecomposable module or zero.